2001 Fiscal Year Final Research Report Summary
Association schemes and spin models
| Project/Area Number |
10440011
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | KYUSHU UNIVERSITY |
|---|
Principal Investigator |
BANNNAI Etsuko Kyushu Univ., Graduate School of Math., Assoc. Professor, 大学院・数理研究院, 助教授 (00253394)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
ITO Tatsuro Kanazawa Univ., Faculty of Science, Professor, 理学部, 教授 (90015909)
MUNEMASA Akihiro Kyushu Univ., Graduate School of Math., Assoc. Professor, 大学院・数理研究院, 助教授 (50219862)
BANNAI Etsuko Kyushu Univ., Graduate School of Math., Professor, 大学院・数理研究院, 教授 (10011652)
NOMURA Kazumasa Tokyo Medical and Dental Univ., Faculty of Cultural Science, Professor, 教養部, 教授 (40111645)
|
|---|
| Project Period (FY) |
1998 – 2001
|
| Keywords | Spin model / Association Schemes / designs / Bose-Mesnsr algebra / type II matrices / s-distance sets / modular invariance / 4-weight spin models |
|---|
| Research Abstract |
The main purpose of this research was to find out sets of points with finite cardinality with a "good" configuration through investigation of relations between spin models and association schemes. Association scheme is the one of the most important objects in the study of algebraic combinatorics. On the other hand spin models give topological invariants of the knots and links in the 3 dimensional Euclidean space R3. During the period we received this grant we obtained the following results. (1) We found that Bose-Mesner algebras attached to a 4-weight spin model coincide with a unique Bose-Mesner algebras. (2) We tried to classify 4-weight spin models with small size and obtained some partial results. (3) We found out that the existence of a 4-weight spin model with exactly two valued on W_2 is equivalent to the existence of a symmetric design with some polarity. (4) We defined type II codes on finite abelian groups using the solutions of modular invariance equations of finite abelian group association schemes. The solutions of this modular invariance equations are known to give spin models. (5) We found out that a upper bound for the cardinality of an s-distance set in Euclidean spaces coincides with the lower bound for the cardinality of 2s-design given by Delsarte-Seidel. In particular we found out that if we assume that s-distance set is antipodal, then a upper bound coincides with the lower bound for the cardinality of antipodal 2s - 1-design given by Delsarte-Seidel. However the situation is different from the spherical case.
The grant we received was mainly used for the travel expenses of us and also the re-searchers in Japan or oversea who are working on related subject with our research. We could make important discussions with many researchers in related topics. We could organized "Mini Conference on Algebraic Combinatorics" at Kyushu University 4 times.
|
